How to make one object fit into volume of another?
How about:


Make a box
Make a circle which fits the box
Make a Lattice object which fits the box
Assign a Lattice modifier to the circle and the box


Now, whatever you do to the lattice, will happen to the circle and the box. 



Or, if you have to fit to an existing box, make the Lattice to fit the circle, and then snap its vertices to the box.